Robotics. Dynamics. Marc Toussaint U Stuttgart


 Milo Greer
 1 years ago
 Views:
Transcription
1 Robotics Dynamics 1D point mass, damping & oscillation, PID, dynamics of mechanical systems, EulerLagrange equation, NewtonEuler recursion, general robot dynamics, joint space control, reference trajectory following, operational space control Marc Toussaint U Stuttgart
2 Kinematic Dynamic instantly change joint velocities q: instantly change joint torques u: δq t! = J (y φ(q t )) u! =? accounts for kinematic coupling of joints but ignores inertia, forces, torques gears, stiff, all of industrial robots accounts for dynamic coupling of joints and full Newtonian physics future robots, compliant, few research robots 2/36
3 When velocities cannot be changed/set arbitrarily Examples: An air plane flying: You cannot command it to hold still in the air, or to move straight up. A car: you cannot command it to move sidewards. Your arm: you cannot command it to throw a ball with arbitrary speed (force limits). A torque controlled robot: You cannot command it to instantly change velocity (infinite acceleration/torque). What all examples have in comment: One can set controls u t (air plane s control stick, car s steering wheel, your muscles activations, torque/voltage/current send to a robot s motors) But these controls only indirectly influence the dynamics of state, x t+1 = f(x t, u t ) 3/36
4 Dynamics The dynamics of a system describes how the controls u t influence the changeofstate of the system x t+1 = f(x t, u t ) The notation x t refers to the dynamic state of the system: e.g., joint positions and velocities x t = (q t, q t ). f is an arbitrary function, often smooth 4/36
5 Outline We start by discussing a 1D point mass for 3 reasons: The most basic forcecontrolled system with inertia We can introduce and understand PID control The behavior of a point mass under PID control is a reference that we can also follow with arbitrary dynamic robots (if the dynamics are known) We discuss computing the dynamics of general robotic systems EulerLagrange equations EulerNewton method We derive the dynamic equivalent of inverse kinematics: operational space control 5/36
6 PID and a 1D point mass 6/36
7 The dynamics of a 1D point mass Start with simplest possible example: 1D point mass (no gravity, no friction, just a single mass) The state x(t) = (q(t), q(t)) is described by: position q(t) R velocity q(t) R The controls u(t) is the force we apply on the mass point The system dynamics is: q(t) = u(t)/m 7/36
8 1D point mass proportional feedback Assume current position is q. The goal is to move it to the position q. What can we do? 8/36
9 1D point mass proportional feedback Assume current position is q. The goal is to move it to the position q. What can we do? Idea 1: Always pull the mass towards the goal q : u = K p (q q) 8/36
10 1D point mass proportional feedback What s the effect of this control law? m q = u = K p (q q) q = q(t) is a function of time, this is a second order differential equation 9/36
11 1D point mass proportional feedback What s the effect of this control law? m q = u = K p (q q) q = q(t) is a function of time, this is a second order differential equation Solution: assume q(t) = a + be ωt (a nonimaginary alternative would be q(t) = a + b ɛ λt cos(ωt)) 9/36
12 1D point mass proportional feedback What s the effect of this control law? m q = u = K p (q q) q = q(t) is a function of time, this is a second order differential equation Solution: assume q(t) = a + be ωt (a nonimaginary alternative would be q(t) = a + b ɛ λt cos(ωt)) m b ω 2 e ωt = K p q K p a K p b e ωt (m b ω 2 + K p b) e ωt = K p (q a) (m b ω 2 + K p b) = 0 (q a) = 0 ω = i K p /m q(t) = q + b e i K p/m t This is an oscillation around q with amplitude b = q(0) q and frequency K p /m! 9/36
13 1D point mass proportional feedback m q = u = K p (q q) q(t) = q + b e i K p/m t Oscillation around q with amplitude b = q(0) q and frequency Kp /m 1 cos(x) /36
14 1D point mass derivative feedback Idea 2 Pull less, when we re heading the right direction already: Damp the system: u = K p (q q) + K d ( q q) q is a desired goal velocity For simplicity we set q = 0 in the following. 11/36
15 1D point mass derivative feedback What s the effect of this control law? m q = u = K p (q q) + K d (0 q) Solution: again assume q(t) = a + be ωt m b ω 2 e ωt = K p q K p a K p b e ωt K d b ωe ωt (m b ω 2 + K d b ω + K p b) e ωt = K p (q a) (m ω 2 + K d ω + K p ) = 0 (q a) = 0 ω = K d ± K 2 d 4mK p 2m q(t) = q + b e ω t The term K d 2m in ω is real exponential decay (damping) 12/36
16 1D point mass derivative feedback q(t) = q + b e ω t, ω = K d ± K 2 d 4mK p 2m Effect of the second term K 2 d 4mK p/2m in ω: K 2 d < 4mK p ω has imaginary part oscillating with frequency K p /m K 2 d /4m2 Kd 2 > 4mK p ω real strongly damped q(t) = q + be K d/2m t e i K p/m K 2 d /4m2 t Kd 2 = 4mK p second term zero only exponential decay 13/36
17 1D point mass derivative feedback illustration from O. Brock s lecture 14/36
18 1D point mass derivative feedback Alternative parameterization: Instead of the gains K p and K d it is sometimes more intuitive to set the wave length λ = 1 1 ω 0 =, K p = m/λ 2 Kp/m damping ratio ξ = ξ > 1: overdamped ξ = 1: critically dampled ξ < 1: oscillatorydamped K d = λk d 4mKp 2m, K d = 2mξ/λ q(t) = q + be ξ t/λ e i 1 ξ 2 t/λ 15/36
19 1D point mass integral feedback Idea 3 Pull if the position error accumulated large in the past: u = K p (q q) + K d ( q q) + K i t s=0 (q (s) q(s)) ds This is not a linear ODE w.r.t. x = (q, q). However, when we extend the state to x = (q, q, e) we have the ODE q = q q = u/m = K p/m(q q) + K d /m( q q) + K i /m e ė = q q (no explicit discussion here) 16/36
20 1D point mass PID control u = K p (q q) + K d ( q q) + K i t s=0 (q q(s)) ds PID control Proportional Control ( Position Control ) f K p (q q) Derivative Control ( Damping ) f K d ( q q) (ẋ = 0 damping) Integral Control ( Steady State Error ) f K i t s=0 (q (s) q(s)) ds 17/36
21 Controlling a 1D point mass lessons learnt Proportional and derivative feedback (PD control) are like adding a spring and damper to the point mass PD control is a linear control law (q, q) u = K p (q q) + K d ( q q) (linear in the dynamic system state x = (q, q)) With such linear control laws we can design approach trajectories (by tuning the gains) but no optimality principle behind such motions 18/36
22 Dynamics of mechanical systems 19/36
23 Two ways to derive dynamics equations for mechanical systems The EulerLagrange equation d L dt q L q = u Used when you want to derive analytic equations of motion ( on paper ) The NewtonEuler recursion f i = m v i, (and related algorithms) u i = I i ẇ + w Iw Algorithms that propagate forces through a kinematic tree and numerically compute the inverse dynamics u = NE(q, q, q) or forward dynamics q = f(q, q, u). 20/36
24 The EulerLagrange equation d L dt q L q = u L(q, q) is called Lagrangian and defined as L = T U where T =kinetic energy and U =potential energy. q is called generalized coordinate any coordinates such that (q, q) describes the state of the system. Joint angles in our case. u are external forces 21/36
25 The EulerLagrange equation How is this typically done? First, describe the kinematics and Jacobians for every link i: (q, q) {T W i(q), v i, w i} Recall T W i (q) = T W A T A A (q) T A B T B B (q) Further, we know that a link s velocity v i = J i q can be described via its position Jacobian. Similarly we can describe the link s angular velocity w i = Ji w q as linear in q. Second, formulate the kinetic energy T = i 1 2 miv2 i w i I iw i = i 1 2 q M i q, M i = J i Ji w m ii I i where I i = R i Ī ir i and Īi the inertia tensor in link coordinates Third, formulate the potential energies (typically independent of q) U = gm iheight(i) Fourth, compute the partial derivatives analytically to get something like }{{} u = d L dt q L q = }{{} M control inertia q + Ṁ q T q } {{ } Coriolis + U q }{{} gravity J i Ji w which relates accelerations q to the forces 22/36
26 Example: A pendulum Generalized coordinates: angle q = (θ) Kinematics: velocity of the mass: v = (l θ cos θ, 0, l θ sin θ) angular velocity of the mass: w = (0, θ, 0) Energies: T = 1 2 mv w Iw = 1 2 (ml2 + I 2 ) θ 2, U = mgl cos θ EulerLagrange equation: u = d L dt q L q = d dt (ml2 + I 2 ) θ + mgl sin θ = (ml 2 + I 2 ) θ + mgl sin θ 23/36
27 NewtonEuler recursion An algorithms that compute the inverse dynamics u = NE(q, q, q ) by recursively computing force balance at each joint: Newton s equation expresses the force acting at the center of mass for an accelerated body: f i = m v i Euler s equation expresses the torque (=control!) acting on a rigid body given an angular velocity and angular acceleration: u i = I iẇ + w Iw Forward recursion: ( kinematics) Compute (angular) velocities (v i, w i ) and accelerations ( v i, ẇ i ) for every link (via forward propagation; see geometry notes for details) Backward recursion: For the leaf links, we now know the desired accelerations q and can compute the necessary joint torques. Recurse backward. 24/36
28 Numeric algorithms for forward and inverse dynamics NewtonEuler recursion: very fast (O(n)) method to compute inverse dynamics u = NE(q, q, q ) Note that we can use this algorithm to also compute gravity terms: u = NE(q, 0, 0) = G(q) Coriolis terms: u = NE(q, q, 0) = C(q, q) q column of Intertia matrix: u = NE(q, 0, e i) = M(q) e i ArticulatedBodyDynamics: fast method (O(n)) to compute forward dynamics q = f(q, q, u) 25/36
29 Some last practical comments [demo] Use energy conservation to measure dynamic of physical simulation Physical simulation engines (developed for games): ODE (Open Dynamics Engine) Bullet (originally focussed on collision only) Physx (Nvidia) Differences of these engines to Lagrange, NE or ABD: Game engine can model much more: Contacts, tissues, particles, fog, etc (The way they model contacts looks ok but is somewhat fictional) On kinematic trees, NE or ABD are much more precise than game engines Game engines do not provide inverse dynamics, u = NE(q, q, q) Proper modelling of contacts is really really hard 26/36
30 Dynamic control of a robot 27/36
31 We previously learnt the effect of PID control on a 1D point mass Robots are not a 1D point mass Neither is each joint a 1D point mass Applying separate PD control in each joint neglects force coupling (Poor solution: Apply very high gains separately in each joint make joints stiff, as with gears.) However, knowing the robot dynamics we can transfer our understanding of PID control of a point mass to general systems 28/36
32 General robot dynamics Let (q, q) be the dynamic state and u R n the controls (typically joint torques in each motor) of a robot Robot dynamics can generally be written as: M(q) R n n C(q, q) R n G(q) R n M(q) q + C(q, q) q + G(q) = u is positive definite intertia matrix (can be inverted forward simulation of dynamics) are the centripetal and coriolis forces are the gravitational forces u are the joint torques (cf. to the EulerLagrange equation on slide 22) We often write more compactly: M(q) q + F (q, q) = u 29/36
33 Controlling a general robot From now on we jsut assume that we have algorithms to efficiently compute M(q) and F (q, q) for any (q, q) Inverse dynamics: If we know the desired q for each joint, gives the necessary torques u = M(q) q + F (q, q) Forward dynamics: If we know which torques u we apply, use q = M(q) 1 (u F (q, q)) to simulate the dynamics of the system (e.g., using RungeKutta) 30/36
34 Controlling a general robot joint space approach Where could we get the desired q from? Assume we have a nice smooth reference trajectory q ref 0:T (generated with some motion profile or alike), we can at each t read off the desired acceleration as q ref t := 1 τ [(q t+1 q t )/τ (q t q t1 )/τ] = (q t1 + q t+1 2q t )/τ 2 However, tiny errors in acceleration will accumulate greatly over time! This is Instable!! 31/36
35 Controlling a general robot joint space approach Where could we get the desired q from? Assume we have a nice smooth reference trajectory q ref 0:T (generated with some motion profile or alike), we can at each t read off the desired acceleration as q ref t := 1 τ [(q t+1 q t )/τ (q t q t1 )/τ] = (q t1 + q t+1 2q t )/τ 2 However, tiny errors in acceleration will accumulate greatly over time! This is Instable!! Choose a desired acceleration q t that implies a PDlike behavior around the reference trajectory! q t = q ref t + K p (q ref t q t ) + K d ( q ref t q t ) This is a standard and very convenient heuristic to track a reference trajectory when the robot dynamics are known: All joints will exactly behave like a 1D point particle around the reference trajectory! 31/36
36 Controlling a robot operational space approach Recall the inverse kinematics problem: We know the desired step δy (or velocity ẏ ) of the endeffector. Which step δq (or velocities q) should we make in the joints? Equivalent dynamic problem: We know how the desired acceleration ÿ of the endeffector. What controls u should we apply? 32/36
37 Operational space control Inverse kinematics: q = argmin q φ(q) y 2 C + q q 0 2 W Operational space control (one might call it Inverse task space dynamics ): u = argmin φ(q) ÿ 2 C + u 2 H u 33/36
38 Operational space control We can derive the optimum perfectly analogous to inverse kinematics We identify the minimum of a locally squared potential, using the local linearization (and approx. J = 0) We get φ(q) = d dt φ(q) d (J q + Jq) J q + 2J dt q = JM 1 (u F ) + 2J q u = T (ÿ 2 J q + T F ) with T = JM 1, (C T = H 1 T (T H 1 T ) 1 ) T = (T CT + H) 1 T C 34/36
39 Controlling a robot operational space approach Where could we get the desired ÿ from? Reference trajectory y0:t ref in operational space PDlike behavior in each operational space: ÿt = ÿt ref + K p (yt ref y t ) + K d (ẏt ref ẏ t ) illustration from O. Brock s lecture Operational space control: Let the system behave as if we could directly apply a 1D point mass behavior to the endeffector 35/36
40 Multiple tasks Recall trick last time: we defined a big kinematic map Φ(q) such that q = argmin q q q 0 2 W + Φ(q) 2 Works analogously in the dynamic case: u = argmin u u 2 H + Φ(q) 2 36/36
Robotics. Dynamics. University of Stuttgart Winter 2018/19
Robotics Dynamics 1D point mass, damping & oscillation, PID, dynamics of mechanical systems, EulerLagrange equation, NewtonEuler, joint space control, reference trajectory following, optimal operational
More informationIntroduction to Robotics
Introduction to Robotics Marc Toussaint April 2016 This is a direct concatenation and reformatting of all lecture slides and exercises from the Robotics course (winter term 2014/15, U Stuttgart), including
More informationRobotics & Automation. Lecture 25. Dynamics of Constrained Systems, Dynamic Control. John T. Wen. April 26, 2007
Robotics & Automation Lecture 25 Dynamics of Constrained Systems, Dynamic Control John T. Wen April 26, 2007 Last Time Order N Forward Dynamics (3sweep algorithm) Factorization perspective: causalanticausal
More informationRobotics. Control Theory. Marc Toussaint U Stuttgart
Robotics Control Theory Topics in control theory, optimal control, HJB equation, infinite horizon case, LinearQuadratic optimal control, Riccati equations (differential, algebraic, discretetime), controllability,
More informationq 1 F m d p q 2 Figure 1: An automated crane with the relevant kinematic and dynamic definitions.
Robotics II March 7, 018 Exercise 1 An automated crane can be seen as a mechanical system with two degrees of freedom that moves along a horizontal rail subject to the actuation force F, and that transports
More informationClassical Mechanics Comprehensive Exam Solution
Classical Mechanics Comprehensive Exam Solution January 31, 011, 1:00 pm 5:pm Solve the following six problems. In the following problems, e x, e y, and e z are unit vectors in the x, y, and z directions,
More informationPhysics 106a, Caltech 4 December, Lecture 18: Examples on Rigid Body Dynamics. Rotating rectangle. Heavy symmetric top
Physics 106a, Caltech 4 December, 2018 Lecture 18: Examples on Rigid Body Dynamics I go through a number of examples illustrating the methods of solving rigid body dynamics. In most cases, the problem
More informationCase Study: The Pelican Prototype Robot
5 Case Study: The Pelican Prototype Robot The purpose of this chapter is twofold: first, to present in detail the model of the experimental robot arm of the Robotics lab. from the CICESE Research Center,
More informationMultibody simulation
Multibody simulation Dynamics of a multibody system (EulerLagrange formulation) Dimitar Dimitrov Örebro University June 16, 2012 Main points covered EulerLagrange formulation manipulator inertia matrix
More informationDynamics. Basilio Bona. Semester 1, DAUIN Politecnico di Torino. B. Bona (DAUIN) Dynamics Semester 1, / 18
Dynamics Basilio Bona DAUIN Politecnico di Torino Semester 1, 201617 B. Bona (DAUIN) Dynamics Semester 1, 201617 1 / 18 Dynamics Dynamics studies the relations between the 3D space generalized forces
More informationIntroduction to Robotics
J. Zhang, L. Einig 277 / 307 MIN Faculty Department of Informatics Lecture 8 Jianwei Zhang, Lasse Einig [zhang, einig]@informatik.unihamburg.de University of Hamburg Faculty of Mathematics, Informatics
More informationLecture «Robot Dynamics»: Dynamics and Control
Lecture «Robot Dynamics»: Dynamics and Control 151085100 V lecture: CAB G11 Tuesday 10:15 12:00, every week exercise: HG E1.2 Wednesday 8:15 10:00, according to schedule (about every 2nd week) Marco
More informationArtificial Intelligence & Neuro Cognitive Systems Fakultät für Informatik. Robot Dynamics. Dr.Ing. John Nassour J.
Artificial Intelligence & Neuro Cognitive Systems Fakultät für Informatik Robot Dynamics Dr.Ing. John Nassour 25.1.218 J.Nassour 1 Introduction Dynamics concerns the motion of bodies Includes Kinematics
More informationIn this section of notes, we look at the calculation of forces and torques for a manipulator in two settings:
Introduction Up to this point we have considered only the kinematics of a manipulator. That is, only the specification of motion without regard to the forces and torques required to cause motion In this
More information2007 Problem Topic Comment 1 Kinematics Positiontime equation Kinematics 7 2 Kinematics Velocitytime graph Dynamics 6 3 Kinematics Average velocity
2007 Problem Topic Comment 1 Kinematics Positiontime equation Kinematics 7 2 Kinematics Velocitytime graph Dynamics 6 3 Kinematics Average velocity Energy 7 4 Kinematics Free fall Collisions 3 5 Dynamics
More informationLecture Note 12: Dynamics of Open Chains: Lagrangian Formulation
ECE5463: Introduction to Robotics Lecture Note 12: Dynamics of Open Chains: Lagrangian Formulation Prof. Wei Zhang Department of Electrical and Computer Engineering Ohio State University Columbus, Ohio,
More informationPhysics 8, Fall 2011, equation sheet work in progress
1 year 3.16 10 7 s Physics 8, Fall 2011, equation sheet work in progress circumference of earth 40 10 6 m speed of light c = 2.9979 10 8 m/s mass of proton or neutron 1 amu ( atomic mass unit ) = 1 1.66
More informationLecture «Robot Dynamics»: Dynamics 2
Lecture «Robot Dynamics»: Dynamics 2 151085100 V lecture: CAB G11 Tuesday 10:15 12:00, every week exercise: HG E1.2 Wednesday 8:15 10:00, according to schedule (about every 2nd week) office hour: LEE
More informationLecture 9: Eigenvalues and Eigenvectors in Classical Mechanics (See Section 3.12 in Boas)
Lecture 9: Eigenvalues and Eigenvectors in Classical Mechanics (See Section 3 in Boas) As suggested in Lecture 8 the formalism of eigenvalues/eigenvectors has many applications in physics, especially in
More informationMultibody simulation
Multibody simulation Dynamics of a multibody system (NewtonEuler formulation) Dimitar Dimitrov Örebro University June 8, 2012 Main points covered NewtonEuler formulation forward dynamics inverse dynamics
More informationOscillations. Phys101 Lectures 28, 29. Key points: Simple Harmonic Motion (SHM) SHM Related to Uniform Circular Motion The Simple Pendulum
Phys101 Lectures 8, 9 Oscillations Key points: Simple Harmonic Motion (SHM) SHM Related to Uniform Circular Motion The Simple Pendulum Ref: 111,,3,4. Page 1 Oscillations of a Spring If an object oscillates
More informationChapter 14. Oscillations. Oscillations Introductory Terminology Simple Harmonic Motion:
Chapter 14 Oscillations Oscillations Introductory Terminology Simple Harmonic Motion: Kinematics Energy Examples of Simple Harmonic Oscillators Damped and Forced Oscillations. Resonance. Periodic Motion
More informationDistance travelled time taken and if the particle is a distance s(t) along the xaxis, then its instantaneous speed is:
Chapter 1 Kinematics 1.1 Basic ideas r(t) is the position of a particle; r = r is the distance to the origin. If r = x i + y j + z k = (x, y, z), then r = r = x 2 + y 2 + z 2. v(t) is the velocity; v =
More informationDynamics. describe the relationship between the joint actuator torques and the motion of the structure important role for
Dynamics describe the relationship between the joint actuator torques and the motion of the structure important role for simulation of motion (test control strategies) analysis of manipulator structures
More informationLagrangian Dynamics: Generalized Coordinates and Forces
Lecture Outline 1 2.003J/1.053J Dynamics and Control I, Spring 2007 Professor Sanjay Sarma 4/2/2007 Lecture 13 Lagrangian Dynamics: Generalized Coordinates and Forces Lecture Outline Solve one problem
More information3 Space curvilinear motion, motion in noninertial frames
3 Space curvilinear motion, motion in noninertial frames 3.1 Inclass problem A rocket of initial mass m i is fired vertically up from earth and accelerates until its fuel is exhausted. The residual mass
More informationRobotics. Path Planning. Marc Toussaint U Stuttgart
Robotics Path Planning Path finding vs. trajectory optimization, local vs. global, Dijkstra, Probabilistic Roadmaps, Rapidly Exploring Random Trees, nonholonomic systems, car system equation, pathfinding
More informationReview: control, feedback, etc. Today s topic: statespace models of systems; linearization
Plan of the Lecture Review: control, feedback, etc Today s topic: statespace models of systems; linearization Goal: a general framework that encompasses all examples of interest Once we have mastered
More informationChaotic motion. Phys 420/580 Lecture 10
Chaotic motion Phys 420/580 Lecture 10 Finitedifference equations Finite difference equation approximates a differential equation as an iterative map (x n+1,v n+1 )=M[(x n,v n )] Evolution from time t
More informationChaotic motion. Phys 420/580 Lecture 10
Chaotic motion Phys 420/580 Lecture 10 Finitedifference equations Finite difference equation approximates a differential equation as an iterative map (x n+1,v n+1 )=M[(x n,v n )] Evolution from time t
More informationPhysics 8, Fall 2013, equation sheet work in progress
(Chapter 1: foundations) 1 year 3.16 10 7 s Physics 8, Fall 2013, equation sheet work in progress circumference of earth 40 10 6 m speed of light c = 2.9979 10 8 m/s mass of proton or neutron 1 amu ( atomic
More informationChaotic motion. Phys 750 Lecture 9
Chaotic motion Phys 750 Lecture 9 Finitedifference equations Finite difference equation approximates a differential equation as an iterative map (x n+1,v n+1 )=M[(x n,v n )] Evolution from time t =0to
More informationCP1 REVISION LECTURE 3 INTRODUCTION TO CLASSICAL MECHANICS. Prof. N. Harnew University of Oxford TT 2017
CP1 REVISION LECTURE 3 INTRODUCTION TO CLASSICAL MECHANICS Prof. N. Harnew University of Oxford TT 2017 1 OUTLINE : CP1 REVISION LECTURE 3 : INTRODUCTION TO CLASSICAL MECHANICS 1. Angular velocity and
More informationArticulated body dynamics
Articulated rigid bodies Articulated body dynamics Beyond human models How would you represent a pose? Quadraped animals Wavy hair Animal fur Plants Maximal vs. reduced coordinates How are things connected?
More information28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod)
28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod) θ + ω 2 sin θ = 0. Indicate the stable equilibrium points as well as the unstable equilibrium points.
More informationVideo 8.1 Vijay Kumar. Property of University of Pennsylvania, Vijay Kumar
Video 8.1 Vijay Kumar 1 Definitions State State equations Equilibrium 2 Stability Stable Unstable Neutrally (Critically) Stable 3 Stability Translate the origin to x e x(t) =0 is stable (Lyapunov stable)
More informationIn the presence of viscous damping, a more generalized form of the Lagrange s equation of motion can be written as
2 MODELING Once the control target is identified, which includes the state variable to be controlled (ex. speed, position, temperature, flow rate, etc), and once the system drives are identified (ex. force,
More informationSimple and Physical Pendulums Challenge Problem Solutions
Simple and Physical Pendulums Challenge Problem Solutions Problem 1 Solutions: For this problem, the answers to parts a) through d) will rely on an analysis of the pendulum motion. There are two conventional
More informationRobot Control Basics CS 685
Robot Control Basics CS 685 Control basics Use some concepts from control theory to understand and learn how to control robots Control Theory general field studies control and understanding of behavior
More informationAdaptive Robust Tracking Control of Robot Manipulators in the Taskspace under Uncertainties
Australian Journal of Basic and Applied Sciences, 3(1): 308322, 2009 ISSN 19918178 Adaptive Robust Tracking Control of Robot Manipulators in the Taskspace under Uncertainties M.R.Soltanpour, M.M.Fateh
More informationRobot Dynamics II: Trajectories & Motion
Robot Dynamics II: Trajectories & Motion Are We There Yet? METR 4202: Advanced Control & Robotics Dr Surya Singh Lecture # 5 August 23, 2013 metr4202@itee.uq.edu.au http://itee.uq.edu.au/~metr4202/ 2013
More informationManipulator Dynamics 2. Instructor: Jacob Rosen Advanced Robotic  MAE 263D  Department of Mechanical & Aerospace Engineering  UCLA
Manipulator Dynamics 2 Forward Dynamics Problem Given: Joint torques and links geometry, mass, inertia, friction Compute: Angular acceleration of the links (solve differential equations) Solution Dynamic
More informationDynamics and Time Integration. Computer Graphics CMU /15662, Fall 2016
Dynamics and Time Integration Computer Graphics CMU 15462/15662, Fall 2016 Last Quiz: IK (and demo) Last few classes Keyframing and motion capture  Q: what can one NOT do with these techniques? The
More informationLagrange s Equations of Motion and the Generalized Inertia
Lagrange s Equations of Motion and the Generalized Inertia The Generalized Inertia Consider the kinetic energy for a n degree of freedom mechanical system with coordinates q, q 2,... q n. If the system
More informationChapter 12. Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = kx
Chapter 1 Lecture Notes Chapter 1 Oscillatory Motion Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = kx When the mass is released, the spring will pull
More informationModeling and Experimentation: Compound Pendulum
Modeling and Experimentation: Compound Pendulum Prof. R.G. Longoria Department of Mechanical Engineering The University of Texas at Austin Fall 2014 Overview This lab focuses on developing a mathematical
More informationProblem 1: Lagrangians and Conserved Quantities. Consider the following action for a particle of mass m moving in one dimension
105A Practice Final Solutions March 13, 01 William Kelly Problem 1: Lagrangians and Conserved Quantities Consider the following action for a particle of mass m moving in one dimension S = dtl = mc dt 1
More information1 Trajectory Generation
CS 685 notes, J. Košecká 1 Trajectory Generation The material for these notes has been adopted from: John J. Craig: Robotics: Mechanics and Control. This example assumes that we have a starting position
More informationAdvanced Robotic Manipulation
Advanced Robotic Manipulation Handout CS37A (Spring 017 Solution Set # Problem 1  Redundant robot control The goal of this problem is to familiarize you with the control of a robot that is redundant with
More informationRobotics. Kinematics. Marc Toussaint University of Stuttgart Winter 2017/18
Robotics Kinematics 3D geometry, homogeneous transformations, kinematic map, Jacobian, inverse kinematics as optimization problem, motion profiles, trajectory interpolation, multiple simultaneous tasks,
More informationMidterm 3 Review (Ch 914)
Midterm 3 Review (Ch 914) PowerPoint Lectures for University Physics, Twelfth Edition Hugh D. Young and Roger A. Freedman Lectures by James Pazun Copyright 2008 Pearson Education Inc., publishing as Pearson
More informationContents. Dynamics and control of mechanical systems. Focus on
Dynamics and control of mechanical systems Date Day 1 (01/08) Day 2 (03/08) Day 3 (05/08) Day 4 (07/08) Day 5 (09/08) Day 6 (11/08) Content Review of the basics of mechanics. Kinematics of rigid bodies
More informationSymmetries 2  Rotations in Space
Symmetries 2  Rotations in Space This symmetry is about the isotropy of space, i.e. space is the same in all orientations. Thus, if we continuously rotated an entire system in space, we expect the system
More informationClassical Mechanics. FIG. 1. Figure for (a), (b) and (c). FIG. 2. Figure for (d) and (e).
Classical Mechanics 1. Consider a cylindrically symmetric object with a total mass M and a finite radius R from the axis of symmetry as in the FIG. 1. FIG. 1. Figure for (a), (b) and (c). (a) Show that
More informationSimple Car Dynamics. Outline. Claude Lacoursière HPC2N/VRlab, Umeå Universitet, Sweden, May 18, 2005
Simple Car Dynamics Claude Lacoursière HPC2N/VRlab, Umeå Universitet, Sweden, and CMLabs Simulations, Montréal, Canada May 18, 2005 Typeset by FoilTEX May 16th 2005 Outline basics of vehicle dynamics different
More informationTrajectorytracking control of a planar 3RRR parallel manipulator
Trajectorytracking control of a planar 3RRR parallel manipulator Chaman Nasa and Sandipan Bandyopadhyay Department of Engineering Design Indian Institute of Technology Madras Chennai, India Abstract
More informationLinearize a nonlinear system at an appropriately chosen point to derive an LTI system with A, B,C, D matrices
Dr. J. Tani, Prof. Dr. E. Frazzoli 151059100 Control Systems I (HS 2018) Exercise Set 2 Topic: Modeling, Linearization Discussion: 5. 10. 2018 Learning objectives: The student can mousavis@ethz.ch, 4th
More informationRotation. Kinematics Rigid Bodies Kinetic Energy. Torque Rolling. featuring moments of Inertia
Rotation Kinematics Rigid Bodies Kinetic Energy featuring moments of Inertia Torque Rolling Angular Motion We think about rotation in the same basic way we do about linear motion How far does it go? How
More informationAssignments VIII and IX, PHYS 301 (Classical Mechanics) Spring 2014 Due 3/21/14 at start of class
Assignments VIII and IX, PHYS 301 (Classical Mechanics) Spring 2014 Due 3/21/14 at start of class Homeworks VIII and IX both center on Lagrangian mechanics and involve many of the same skills. Therefore,
More informationRotational motion problems
Rotational motion problems. (Massive pulley) Masses m and m 2 are connected by a string that runs over a pulley of radius R and moment of inertia I. Find the acceleration of the two masses, as well as
More informationSimple Harmonic Motion
Physics 7B1 (A/B) Professor Cebra Winter 010 Lecture 10 Simple Harmonic Motion Slide 1 of 0 Announcements Final exam will be next Wednesday 3:305:30 A Formula sheet will be provided Closednotes & closedbooks
More informationOscillatory Motion. Simple pendulum: linear Hooke s Law restoring force for small angular deviations. small angle approximation. Oscillatory solution
Oscillatory Motion Simple pendulum: linear Hooke s Law restoring force for small angular deviations d 2 θ dt 2 = g l θ small angle approximation θ l Oscillatory solution θ(t) =θ 0 sin(ωt + φ) F with characteristic
More informationTheoretical physics. Deterministic chaos in classical physics. Martin Scholtz
Theoretical physics Deterministic chaos in classical physics Martin Scholtz scholtzzz@gmail.com Fundamental physical theories and role of classical mechanics. Intuitive characteristics of chaos. Newton
More informationChapter 14 Periodic Motion
Chapter 14 Periodic Motion 1 Describing Oscillation First, we want to describe the kinematical and dynamical quantities associated with Simple Harmonic Motion (SHM), for example, x, v x, a x, and F x.
More informationChapter 14 Oscillations. Copyright 2009 Pearson Education, Inc.
Chapter 14 Oscillations Oscillations of a Spring Simple Harmonic Motion Energy in the Simple Harmonic Oscillator Simple Harmonic Motion Related to Uniform Circular Motion The Simple Pendulum The Physical
More informationColumbia University Department of Physics QUALIFYING EXAMINATION
Columbia University Department of Physics QUALIFYING EXAMINATION Monday, January 9, 2017 11:00AM to 1:00PM Classical Physics Section 1. Classical Mechanics Two hours are permitted for the completion of
More informationLecture Schedule Week Date Lecture (M: 2:05p3:50, 50N202)
J = x θ τ = J T F 2018 School of Information Technology and Electrical Engineering at the University of Queensland Lecture Schedule Week Date Lecture (M: 2:05p3:50, 50N202) 1 23Jul Introduction + Representing
More information(W: 12:051:50, 50N202)
2016 School of Information Technology and Electrical Engineering at the University of Queensland Schedule of Events Week Date Lecture (W: 12:051:50, 50N202) 1 27Jul Introduction 2 Representing Position
More informationOscillatory Motion. Simple pendulum: linear Hooke s Law restoring force for small angular deviations. Oscillatory solution
Oscillatory Motion Simple pendulum: linear Hooke s Law restoring force for small angular deviations d 2 θ dt 2 = g l θ θ l Oscillatory solution θ(t) =θ 0 sin(ωt + φ) F with characteristic angular frequency
More informationDamped harmonic motion
Damped harmonic motion March 3, 016 Harmonic motion is studied in the presence of a damping force proportional to the velocity. The complex method is introduced, and the different cases of underdamping,
More informationELEC4631 s Lecture 2: Dynamic Control Systems 7 March Overview of dynamic control systems
ELEC4631 s Lecture 2: Dynamic Control Systems 7 March 2011 Overview of dynamic control systems Goals of Controller design Autonomous dynamic systems Linear Multiinput multioutput (MIMO) systems Bat flight
More informationPHYSICS. Chapter 15 Lecture FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E RANDALL D. KNIGHT Pearson Education, Inc.
PHYSICS FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E Chapter 15 Lecture RANDALL D. KNIGHT Chapter 15 Oscillations IN THIS CHAPTER, you will learn about systems that oscillate in simple harmonic
More informationIn most robotic applications the goal is to find a multibody dynamics description formulated
Chapter 3 Dynamics Mathematical models of a robot s dynamics provide a description of why things move when forces are generated in and applied on the system. They play an important role for both simulation
More informationLecture 13 REVIEW. Physics 106 Spring What should we know? What should we know? Newton s Laws
Lecture 13 REVIEW Physics 106 Spring 2006 http://web.njit.edu/~sirenko/ What should we know? Vectors addition, subtraction, scalar and vector multiplication Trigonometric functions sinθ, cos θ, tan θ,
More informationPhysics 6010, Fall Relevant Sections in Text: Introduction
Physics 6010, Fall 2016 Introduction. Configuration space. Equations of Motion. Velocity Phase Space. Relevant Sections in Text: 1.1 1.4 Introduction This course principally deals with the variational
More informationP321(b), Assignement 1
P31(b), Assignement 1 1 Exercise 3.1 (Fetter and Walecka) a) The problem is that of a point mass rotating along a circle of radius a, rotating with a constant angular velocity Ω. Generally, 3 coordinates
More informationGeneral procedure for formulation of robot dynamics STEP 1 STEP 3. Module 9 : Robot Dynamics & controls
Module 9 : Robot Dynamics & controls Lecture 32 : General procedure for dynamics equation forming and introduction to control Objectives In this course you will learn the following Lagrangian Formulation
More informationLagrangian Dynamics: Derivations of Lagrange s Equations
Constraints and Degrees of Freedom 1.003J/1.053J Dynamics and Control I, Spring 007 Professor Thomas Peacock 4/9/007 Lecture 15 Lagrangian Dynamics: Derivations of Lagrange s Equations Constraints and
More informationPHYSICS 221, FALL 2011 EXAM #2 SOLUTIONS WEDNESDAY, NOVEMBER 2, 2011
PHYSICS 1, FALL 011 EXAM SOLUTIONS WEDNESDAY, NOVEMBER, 011 Note: The unit vectors in the +x, +y, and +z directions of a righthanded Cartesian coordinate system are î, ĵ, and ˆk, respectively. In this
More informationNMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582
NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING NMT EE 589 & UNM ME 482/582 Simplified drive train model of a robot joint Inertia seen by the motor Link k 1 I I D ( q) k mk 2 kk Gk Torque amplification G
More information7 Pendulum. Part II: More complicated situations
MATH 35, by T. Lakoba, University of Vermont 60 7 Pendulum. Part II: More complicated situations In this Lecture, we will pursue two main goals. First, we will take a glimpse at a method of Classical Mechanics
More informationLinearization problem. The simplest example
Linear Systems Lecture 3 1 problem Consider a nonlinear timeinvariant system of the form ( ẋ(t f x(t u(t y(t g ( x(t u(t (1 such that x R n u R m y R p and Slide 1 A: f(xu f(xu g(xu and g(xu exist and
More informationFinal Exam Spring 2014 May 05, 2014
95.141 Final Exam Spring 2014 May 05, 2014 Section number Section instructor Last/First name Last 3 Digits of Student ID Number: Answer all questions, beginning each new question in the space provided.
More informationfor changing independent variables. Most simply for a function f(x) the Legendre transformation f(x) B(s) takes the form B(s) = xs f(x) with s = df
Physics 106a, Caltech 1 November, 2018 Lecture 10: Hamiltonian Mechanics I The Hamiltonian In the Hamiltonian formulation of dynamics each second order ODE given by the Euler Lagrange equation in terms
More informationLecture IV: Time Discretization
Lecture IV: Time Discretization Motivation Kinematics: continuous motion in continuous time Computer simulation: Discrete time steps t Discrete Space (mesh particles) Updating Position Force induces acceleration.
More informationL = 1 2 a(q) q2 V (q).
Physics 3550, Fall 2011 Motion near equilibrium  Small Oscillations Relevant Sections in Text: 5.1 5.6 Motion near equilibrium 1 degree of freedom One of the most important situations in physics is motion
More informationEE Homework 3 Due Date: 03 / 30 / Spring 2015
EE 476  Homework 3 Due Date: 03 / 30 / 2015 Spring 2015 Exercise 1 (10 points). Consider the problem of two pulleys and a mass discussed in class. We solved a version of the problem where the mass was
More informationEN Nonlinear Control and Planning in Robotics Lecture 3: Stability February 4, 2015
EN530.678 Nonlinear Control and Planning in Robotics Lecture 3: Stability February 4, 2015 Prof: Marin Kobilarov 0.1 Model prerequisites Consider ẋ = f(t, x). We will make the following basic assumptions
More information15. Hamiltonian Mechanics
University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 2015 15. Hamiltonian Mechanics Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License
More informationChapter 8: Momentum, Impulse, & Collisions. Newton s second law in terms of momentum:
linear momentum: Chapter 8: Momentum, Impulse, & Collisions Newton s second law in terms of momentum: impulse: Under what SPECIFIC condition is linear momentum conserved? (The answer does not involve collisions.)
More informationMCE 366 System Dynamics, Spring Problem Set 2. Solutions to Set 2
MCE 366 System Dynamics, Spring 2012 Problem Set 2 Reading: Chapter 2, Sections 2.3 and 2.4, Chapter 3, Sections 3.1 and 3.2 Problems: 2.22, 2.24, 2.26, 2.31, 3.4(a, b, d), 3.5 Solutions to Set 2 2.22
More informationVariation Principle in Mechanics
Section 2 Variation Principle in Mechanics Hamilton s Principle: Every mechanical system is characterized by a Lagrangian, L(q i, q i, t) or L(q, q, t) in brief, and the motion of he system is such that
More informationConsider a particle in 1D at position x(t), subject to a force F (x), so that mẍ = F (x). Define the kinetic energy to be.
Chapter 4 Energy and Stability 4.1 Energy in 1D Consider a particle in 1D at position x(t), subject to a force F (x), so that mẍ = F (x). Define the kinetic energy to be T = 1 2 mẋ2 and the potential energy
More information06. Lagrangian Mechanics II
University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 2015 06. Lagrangian Mechanics II Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License
More informationChapter 15+ Revisit Oscillations and Simple Harmonic Motion
Chapter 15+ Revisit Oscillations and Simple Harmonic Motion Revisit: Oscillations Simple harmonic motion ToDo: Pendulum oscillations Derive the parallel axis theorem for moments of inertia and apply it
More informationChapter 14 Oscillations
Chapter 14 Oscillations Chapter Goal: To understand systems that oscillate with simple harmonic motion. Slide 142 Chapter 14 Preview Slide 143 Chapter 14 Preview Slide 144 Chapter 14 Preview Slide 145
More informationPhysics 106b/196b Problem Set 9 Due Jan 19, 2007
Physics 06b/96b Problem Set 9 Due Jan 9, 2007 Version 3: January 8, 2007 This problem set focuses on dynamics in rotating coordinate systems (Section 5.2), with some additional early material on dynamics
More informationModeling and Simulation of the Nonlinear Computed Torque Control in Simulink/MATLAB for an Industrial Robot
Copyright 2013 Tech Science Press SL, vol.10, no.2, pp.95106, 2013 Modeling and Simulation of the Nonlinear Computed Torque Control in Simulink/MATLAB for an Industrial Robot Dǎnuţ Receanu 1 Abstract:
More informationRobot Manipulator Control. Hesheng Wang Dept. of Automation
Robot Manipulator Control Hesheng Wang Dept. of Automation Introduction Industrial robots work based on the teaching/playback scheme Operators teach the task procedure to a robot he robot plays back eecute
More information